14.2 Applications in operations research and economics

Convexity plays a crucial role in operations research and economics. It simplifies optimization problems, ensuring global optimality and enabling efficient algorithms. This mathematical foundation underpins economic theories, modeling consumer preferences and production frontiers.

Linear programming leverages convex geometry, viewing feasible regions as convex polytopes. The simplex method navigates these polytopes to find optimal solutions. Convex optimization applications span resource allocation, portfolio management, and machine learning, showcasing its versatility in real-world problem-solving.

Convexity in Operations Research and Economics

Convexity in optimization problems

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• Convexity ensures global optimality simplifies search for optimal solutions guarantees local optima are also global optima
• Enables efficient algorithms for solving large-scale problems using interior point methods and gradient descent algorithms
• Provides mathematical foundation for economic theories models consumer preferences and production possibility frontiers
• Allows for duality in optimization problems connecting primal and dual problems through complementary slackness conditions

Linear programming and convex geometry

• Geometric interpretation of linear programs views feasible region as a convex polytope with optimal solution at a vertex
• Simplex method moves along edges of the polytope using pivoting operations to find optimal solution
• Duality theory in linear programming explores weak and strong duality principles
• Karush-Kuhn-Tucker (KKT) conditions establish necessary and sufficient conditions for optimality
• Farkas' lemma fundamental result in linear programming provides alternative theorem for systems of linear inequalities

Applications of convex optimization

• Resource allocation solves network flow problems (transportation networks) and assignment problems (job scheduling)
• Production planning addresses diet problem (nutritional requirements) and blending problems (oil refining)
• Portfolio optimization uses mean-variance analysis to balance risk-return tradeoffs in investment strategies
• Game theory applies convex optimization to find Nash equilibrium in zero-sum games (poker)
• Support vector machines in machine learning create maximum margin classifiers for data classification tasks
• Convex relaxations of non-convex problems employ semidefinite programming relaxations to approximate difficult optimization problems

Convex geometry for economic analysis

• General equilibrium theory uses convexity to prove existence of competitive equilibria and analyze Pareto efficiency
• Convex preferences and utility functions model indifference curves and quasi-concave utility functions in consumer choice theory
• Production possibility frontiers represent convex production sets to study efficient allocation of resources (labor, capital)
• Welfare economics applies convexity in first and second fundamental theorems to analyze market efficiency
• Shapley-Folkman theorem explores approximate convexity in large economies with many agents
• Mechanism design utilizes convexity to ensure incentive compatibility and apply the revelation principle in designing economic systems